An Assessment Of Mathematics

An Assessment of Mathematics Asnakew Tagele 1
ORIGINAL ARTICLE
An Assessment of Mathematics Classroom Teaching-
Learning Process: Consistency with Constructivist
Approach
Asnakew Tagele1
Abstract
This study was conducted on 261 students of whom 136 were from Benishangul Gumuz
and 125 from Amhara region attending government secondary schools (from 9-11
graders in the 2005 academic year). They were high achieving students in mathematics
and science selected from different secondary schools in the two regions for the
“talented students” outreach summer program in Bahir Dar University. The objective of
the study was to assess whether the learning classroom environment was compliant with
constructivism. Data about the learning environment in mathematics classroom was
collected using the Constructivist Learning Environment Survey (CLES). The CLES
consists of five dimensions (scales): personal relevance, mathematical uncertainty,
shared control, critical voice, and student negotiation, each scale having six items. In
other words, the instrument contained thirty items that pupils rank via the use of a five-
point Likert scale, ranging from almost never to almost always. One sample t-test was
used to analyze students’ responses. Results showed that the learning environment
(secondary school practices in mathematics classes in Amhara and Benishangul Gumuz
region) was less constructivist compared to the expected average except student
negotiation. That means four out of the five key principles of constructivism were not
sufficiently implemented. However, student negotiation was found to be adequate,
students indicated that enough opportunities existed for them to exchange ideas. Still
students indicated that they were not encouraged to reflect on the viabilities of each
others’ ideas.
Key Words: Constructivism, constructivist learning environment, Amhara region
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1
Lecturer, Department of Psychology, Bahir Dar University, Ethiopia

Ethiop. J. Educ. & Sc. Vol. 12 No 2, March, 2017 2
BACKGROUND OF THE STUDY
The theory about the nature of knowledge,
how human beings learn and the conditions
that best promote learning has undergone a
significant change. As a consequence, there
has been a paradigm shift in approaches to
teaching and learning from behaviorism to
cognitivism and now to constructivism
(Cooper, 1993). Nix, Fraser and Ledbetter
(2005) suggested that constructivism as
both a philosophy and theory of learning
emphasizes knowledge construction than
knowledge transmission or passive
reception. They also indicated that
knowledge construction requires the active
engagement of students. Similarly, Palmer
(2005) indicated that constructivism
presupposes that students actively construct
and modify their own knowledge using
their existing knowledge, beliefs, interests,
and goals to interpret new information or in
response to environmental stimuli.
Since the active nature of students and
placing them at the center of the learning
process is an important element of
constructivism (Cooper, 1993), active
learning method takes its principles from
the educational philosophy of
constructivism (Lynch, 2010). In Ethiopia,
all teachers at all levels of the education
system are required to fully implement
active learning (MOE, 2010). Peer learning
(a one to five student grouping) is also
emphasized being implemented at all levels
of our education system. Applefield, Huber
and Moallen (2001) indicated that peer
learning is underpinned by social
constructivist learning theory. They also
detailed that collaborative peer learning is a
means of preparing learners to become
actively involved in constructing
knowledge for themselves and
understanding how to use it. Hence,
although not explicitly indicated anywhere,
the educational reform in Ethiopia is
grounded on a social constructivist
epistemology.
It is actually evident from the continuous
criticisms of behaviorist approaches to
education while discussing about student
learning. Lecturing is highly discouraged
and instead group discussion is favored.
Moreover, what the government is doing to
improve learning parallels to the conditions
suggested by proponents of the
constructivist learning approaches. It has
been working with promoting student-
centered approach, putting students in
learning peer groups, reducing class size,
supplying textbooks and other materials so
that to make the learning environment
favorable to classroom interaction.
However, the question of how to
implement classroom teaching that is
consistent with a constructivist view of
learning is an issue of concern (Applefield,
et al., 2001). Applefield and his colleagues
indicated that although the perspective of

constructivism has provided educators with
new ways to understand learners and the
nature of learning, translating it into
practice has become to be a considerable
challenge to practitioners. As a novel
conception of learning and learner, the
constructivist perspective, after all, requires
educators to have a sound understanding of
what it means so that to use it
knowledgably and effectively.
The shift to a constructivist approach
requires creating a context that helps
learners to construct knowledge, and/or
modify through negotiation with peers.
Regarding this, constructivists suggested
that effective teaching is creating effective
learning environments where students are
actively participating and engaging with
the material (Applefield, et al., 2001). That
is, for students to actively engage in and
construct meaningful learning,
constructivists suggested that a
constructivist learning environment (a
favorable learning environment) should be
created for them. By constructivist learning
environment, it refers to the classroom
learning environment (including the
physical, social, psychological and
pedagogical contexts) designed based on
the principles of constructivism
(Zualkernan, 2006).
Constructivists claim that students should
find personal relevance in their studies,
share control over their learning, feel free
to express concerns about their learning,
view science as ever changing, and interact
with each other to construct or modify
knowledge (Taylor, Dawson, & Fraser,
1995; Taylor, Fisher, & Fraser, 1997). This
implies that the teacher should make the
subject relevant to the students’ world
outside of school by connecting the subject
with students’ life experiences, engage
them in reflective negotiations with each
other by providing opportunities to explain
and justify their newly developing ideas
and to reflect on the viability of their own
and other students’ ideas, invite them to
share control of the design, management,
and evaluation of their learning, empower
students to express concern about the
quality of teaching and learning activities
and provide opportunities for students to
experience the uncertain nature of
knowledge (Aldridge, Fraser, Taylor &
Chun, 2000).
Borich and Tombari (1995) suggested
that, in a constructivist classroom, the
teacher’s role is to facilitate and guide
students by asking questions that will
lead them to develop their own
conclusions on the subject. Similarly,
Applefield, Huber and Moallen (2001:
51) suggested that the role of the
constructivist teacher is to “stimulate
thinking in learners that result in
meaningful learning, deeper
understanding and transfer of learning to
real world contexts.” Richard (1991) as
cited in Simon (1995) asserted that the
teacher should design and provide tasks
and projects that initiate students to ask
questions, pose problems and set goals.
Simon has indicated that to help
students to become active learners,
teachers should structure plans to guide
exploration and inquiry. He further
suggested that teachers must lead
students through questions and activities
to discover, discuss and verbalize
knowledge.
Applefield, et al., (2001) indicated that
constructivist learning theory posits that
learning always builds upon prior
knowledge and learning is enhanced
when a person sees connection between
current learning and previous
knowledge (including students’ out of
school experiences), potential
implications, applications, and benefits.

They further indicated that in the social
constructivist classroom, students work
primarily in groups and learning is
interactive and dynamic; there is a great
focus and emphasis on social and
communication skills, as well as
collaboration and exchange of ideas.
They therefore recommended that
effective teaching should allow students
to talk and listen, read, write, and reflect
as they approach course content through
problem-solving exercises, simulations,
case studies, role playing, and other
activities-all of which require students
to apply their previous knowledge
and/or what they are currently learning.
Aldridge, Fraser, Taylor and Chen (2000)
conducted a cross national study on 1081
grade 8 and 9 students in Australia and
1879 grade 7 and 8 students in Taiwan
using the constructivist learning
environment survey, a questionnaire with
five scales originally developed by Taylor
and Fraser (1991) to measure students’
perceptions of the extent to which
constructivist approaches are present in
classrooms (the questionnaire has been
explained in the instrumentation section).
Thao-Do, Thi Bac-Ly and Yuenyong (2016)
conducted a similar study on 335 physics
teacher education students in Vietnam.
Except the shared control scale, the mean
score for each of the rest scales in Australia
ranged between sometimes and often. The
range of mean scores in Taiwan follows the
same pattern with that of Australia except
the critical voice scale which is between
seldom and sometimes. The mean scores
for each scale in Vietnam follow a similar
pattern with Taiwan. Although they have
differences, the three countries somehow
emphasize constructivist learning
environment, their classroom learning
environments were consistent with the
constructivists’.
Statement of the Problem
Constructivism assumes that for learning to
occur collaborative social interactions and
context are necessary (Applefield, et al.,
2001). They suggested that social
interaction through questioning and
explaining, challenging and offering timely
support and feedback facilitates
understanding. Thus in order to teach well,
teachers are expected to make the
classroom atmosphere more interactive and
provide contexts; understand the mental
models that students use to perceive the
world and the assumptions they make to
support those models. Could it be possible
to create such classroom learning
environment in our secondary schools?
If teachers in our secondary schools are
practicing the constructivists’ learning
theory, they must invite, encourage and
provoke students to experience the world,
empower them to ask their own questions
and seek their own answers, challenge
them to understand the world’s
complexities, and enquire uncertainty
(Brooks and Brooks, 1999). If that is the
case, the classroom practices should be
consistent with the principles of
constructivism. In other words, teachers
will create a constructivist learning
environment in their classrooms. However,
presumption in this or that way will not be
helpful unless investigation is conducted
about what actually is happening in the
classroom.
The traditional approach (direct instruction
or simply called the lecture method) was
the most widely used instructional strategy
for a long period in our school systems as it
is true throughout the world. Hence, it is
legitimate to presuppose that our teachers
face a problem to escape from the long
lived behaviorist influences and apply
constructivist strategies. It is also worth
mentioning the researcher’s experience that
in the pre-service undergraduate programs,

teachers themselves are not educated in the
constructivist settings and even their
theoretical background about the
perspective is questionable. Moreover,
becoming a teacher who helps students to
search rather than to follow is challenging.
Thus, the extent to which the key principles
of constructivism are applied in the
classroom (the extent to which the
constructivists approach influenced the
everyday classroom practices) need to be
supported by empirical evidences.
Therefore, the main purpose of this study is
to assess the degree to which mathematics
classroom learning environment is
consistent with the key principles of
constructivist approaches to learning in
secondary schools of Benishangul Gumuz
and Amhara Regions. Thus, based on the
arguments above, the following question
was answered at the end of the research
process.
Is the classroom learning
environment in secondary schools of
Amhara and Benishangul Gumuz
Regions consistent with the
constructivist epistemology?
Significance of the Study
While implementing the
constructivist approach to education,
it is a paramount importance to assess
its effectiveness for policy makers,
teachers, students and other educators
so far they are concerned with the
process of education. That is either to
change our educational approaches or
to continue at least by providing help
for teachers and students, this
research will contribute for all
involved in education system of
Ethiopia by creating awareness about
the current status of classroom
learning environment. It will indicate
whether the constructivist theory of
learning is being properly
implemented or practiced. The
findings of this study will make
teachers and students aware of their
educational practices so that they will
challenge their classroom learning
approaches.
METHODOLOGY
Research Design: The purpose of this
study was to assess the consistency of
secondary school mathematics classroom
learning environments with that of the
constructivist learning environment in
Benishangul Gumuz and Amhara regions.
To achieve this objective, mixed methods
approach specifically sequential
explanatory strategy was employed as a
research design. That is both quantitative
and qualitative data were collected
respectively using a questionnaire and an
interview. Qualitative data were collected
after five days from the completion of the
questionnaire for the quantitative ones.
Sample of the Study: This study was
conducted on 261 students attending
summer outreach program in 2013 at Bahir
Dar University. For its outreach program, a
project to encourage “talented students” to
pursue their future study in science and
technology, Bahir Dar University fetched
outperforming (better achieving) students
in mathematics and science from different
secondary schools of both Benishangul
Gumuz and Amhara regions. Of these 261
respondents, 136 were from Benishangul
Gumuz and 125 were from Amhara
Regions attending government secondary
schools (from 9-11 graders in the 2012/13
academic year).
Among them 175 were males and the rest
86 were females. Again, 52 were grade 9,
119 grade 10 and 90 grade 11 students. Six
students were interviewed to further

explain about their ratings. The respondents
were selected based on their academic
achievement, not through random
sampling. However, better achieving
students are believed to be involved in their
learning and to have the required
information about their respective
classroom learning environment. They are
also believed to be reflective about the
classroom experiences than externalizing
failures and responsibilities to teachers
compared to those who achieve lower than
them.
Data Gathering Instrument and Data
Gathering Procedures: Quantitative data
were collected using the revised
Constructivist Learning Environment
Survey (CLES), a questionnaire that
assesses students’ perceptions of their
classroom learning environment. It consists
of five scales each a 5-point scale of
Almost Always, Often, Sometimes, Seldom
and Almost Never, originally developed by
Taylor & Fraser (1991) to monitor the
constructivist approaches to teaching
science and mathematics. The five scales
are "Personal Relevance", "Mathematical
Uncertainty", "Critical Voice", "Shared
Control" and "Student Negotiation". The
30-item questionnaire contains six items
(statements) in each of the five scales about
practices that could take place in a
classroom learning environment. The
questionnaire was translated into Amharic
and pilot tested on two general secondary
schools students of Bahir Dar Town with a
sample of 150 students. The reliabilities of
each scale were modest with no alpha
values less than .5 and each item positively
correlated with other items of the same
scale. Moreover, the reliability of the
overall scale was .76. Concerning validity,
confirmatory factor analysis showed that
items correlate with their respective scales
and each has a factor loading not less than .
3. Qualitative data were also collected
through interview.
The questionnaire was administered face to
face in the Bahir Dar University’s
auditorium. Before respondents start
completing, the researcher explained the
purpose of the questionnaire and assured
them that it will be used for research
purpose. Moreover, they were not required
to indicate their identity. After five days
they completed the scale, six students were
interviewed by the researcher about the
scales and individual items within the
scales informing about the rating results.
The interview data were to substantiate the

responses for the scale. The findings of the
qualitative data are integrated with the
quantitative ones at the discussion section
of the study.
Table 1: Scales of the CLES, and Their Descriptions
Scale Scale Description Sample Item
Personal
Relevance
Extent to which teachers relate
mathematics to students out of school
experiences
I learn about the world
outside of the school
Critical Voice Extent to which students feel that it is
legitimate and beneficial to question
the teachers’ pedagogical plans and
methods
It is OK for me to ask the
teacher ‘why do I have to
learn this?’
Shared Control Extent to which students are invited
to share with the teacher control of
the learning environment.
I help the teacher to plan
what I am going to learn
Student
Negotiation
Extent to which opportunities exist
for students to explain and justify to
other students their newly developing
ideas
I ask other students to
explain their thoughts
Mathematical
Uncertainty
provisional status of Mathematical
knowledge
I learn that mathematics
has changed over time
Data Analysis
Means and standard deviations were
computed to describe the extent of the
emphasis within a classroom learning
environment on (1) making mathematics
seem relevant to the world outside school;
(2) engage students in reflective
negotiations with each other; (3) teachers
inviting students to share control of the
design, management, evaluation of the
learning; (4) students being empowered to
express concern about the quality of
teaching and learning activities; and (5)
students experiencing the uncertain nature
of mathematical knowledge. To identify
the significance of students’ perceptions
about the frequencies of occurrences of the
key aspects of constructivism, the gathered
data were analysed using one sample t-test.
Moreover, data gathered through the
interview have been analysed together with
quantitative results.
RESULTS
The purpose of this study was to assess the
consistency of secondary school
mathematics classroom learning
environments with that of the
constructivists’ in Amhara and Benishangul
Gumuz Regions. That is to evaluate
whether the learning environment was
compliant with constructivism. Hence, the
results of data analysis have been displayed
in the following sequence. First descriptive
statistics for each of the scales are given by
Table 2, next the one sample t-test values
for the scales are presented by Table 3.

Table 2: Descriptive Statistics: Means and Standard Deviations of Scales
Scales N Mean Std. Deviation
Personal Relevance 261 2.26 0.38
Mathematical Uncertainty 261 2.77 0.57
Critical Voice 261 3.06 0.52
Shared Control 261 2.59 0.56
Student Negotiation 261 3.75 0.58
Concerning students’ responses to the
scales, as displayed by Table 2 above, the
least mean score was obtained for the
personal relevance scale (M=2.26,
SD=0.38) indicating that classroom
learning was rarely related to students’ out
of school experience. The next least mean
value was gained for the shared control
scale (mean=2.59) showing that students
were less frequently invited to share control
of the design, management, evaluation of
their learning with their mathematics
teachers.
The means for the uncertainty and critical
voice scales are, 2.79 and 3.06 respectively
which show that students are sometimes
provided with opportunities to experience
the inherent uncertainty and limitations of
mathematical knowledge and they
sometimes critique about the methods and
approaches of their mathematics teachers.
On the other hand, the largest mean value is
found for negotiation scale, 3.75 showing
that students are often provided with the
opportunities to discuss with each other.
The standard deviations for all the scales
are small (ranging from 0.38 to 0.60)
suggesting homogeneity among the ratings
of students.
Table 3: One Sample t-test Values for the Five Scales
Scale Test Value = 3 (Sometimes)
Mean Difference Df t P
Personal Relevance -0.74 260 -31.94 .000
Mathematical Uncertainty -0.23 260 -6.32 .000
Critical Voice 0.06 260 1.96 .051
Shared Control -0.41 260 -11.78 .000
Student Negotiation 0.75 260 20.85 .000
As indicated in Table 3 above, the t-test
values are statistically significant for the
four scales: relevance, uncertainty, shared
control and negotiation scales. However, it
is only for the negotiation scale that the
mean is statistically significantly greater
than the expected mean. The value for the

critical voice scale is nearly statistically
significant.
Major themes (Ideas and Opinions) from
the Qualitative Data
The interview results were summarized
based on the constructivist principles
measured by each of the five scales as
follows.
Personal Relevance
The results from the quantitative data imply
that the contents of mathematics are
perceived to be less relevant for students’
real life. During the interview, students
revealed that the tasks and activities in the
text books are usually abstractions.
Students see little or no real life application
of the complex formulae and proves they
learn.
Shared Control
The quantitative data suggest that students
exercise less shared control in their
mathematics classes. The interview data
confirmed it and revealed that decisions
regarding planning, teaching and
evaluation are made almost exclusively by
teachers. That is, students almost never
experienced shared control in mathematics
subjects. Students indicated that teachers
consult students rarely about what should
be included in their assessment. They are
also very rarely invited in designing
learning activities and that is in preparing
some geometric figures, slide rules and
some other teaching aids. They indicated
that teachers are concerned to cover the
content of the text books and almost never
allow opportunities for them to exercise
control over what they learn.
Mathematical Uncertainty
The result for the quantitative data has been
supported by the qualitative data during the
interview. Students perceived that
mathematics is perfect. They indicated that
the formulae and theorems are proved and
uncertainty doesn’t have a room in
mathematics. The way they learn is to
understand what has been discovered not
the way to discover in mathematics.
Teachers never showed them how axioms,
formulae and theorems are developed and
to try developing their on based on certain
assumption as usually done in mathematics.
Critical Voice
It is not common for students to complain
about the contents taught. They usually
accept what teachers teach since they
consider the contents in the text books
prepared for them and are appropriate.
However, students criticize the relevance of
some mathematics contents to their
classmates though they didn’t directly ask
their teachers. Similarly, students rarely
complain about the teachers’ subject matter
knowledge and their teaching methods.
They sometimes complain to school
principals to change teachers. For example
a teacher teaching grade ten could be
exchanged by another teacher teaching
grade nine. They indicated that though very
rarely students close classroom doors
behind the teacher and resist entering and
teaching them.
Student Negotiation
Compared to other scale values, the
quantitative data for student negotiation
showed that students have better
opportunities. In the interview, students
indicated that their teachers allow
discussing over learning tasks. However,
they indicated that the discussion is on
solving text book problems following
formulae and learned procedures than
negotiating students’ new ideas. Moreover,
teachers allow a brief time and soon start
demonstrating how to solve the problems.
DISCUSSION

In this study, the consistency of secondary
school mathematics classroom learning
environment with that of the constructivist
key principles was investigated. Generally,
students indicated that their mathematics
classrooms sometimes or seldom reflected
constructivist aspects, with the average
item mean ranging from 2.26-3.75 for the
key dimensions (a mean of 3 and 2
corresponds to sometimes and seldom
respectively). The results of the analysis
will be discussed for each of the scales in
the ascending order of their mean values.
The Relevance Scale
This scale was concerned to measure the
extent to which students were provided
with opportunities to relate mathematics
with out- of-school experience.
The analysis revealed that the mean score
was statistically significantly smaller than
the expected mean, (M=2.26),
t(260)=31.94, P<0.001. Compared to
previous studies in Australia, Taiwan and
Vietnam (with mean score 3.17, 3.30 and
3.12 respectively), this mean score is low
(Aldridge, Fraser, Taylor and Chen, 2000;
Thao-Do, Thi Bac-Ly and Yuenyong,
2016).
Students have reported that they have seen
little connection between what they learn in
the classroom and their real life
experiences. This mean value would be
even less than the current value if not
students rated high an item of the scale that
asked them how often they learn the use of
mathematics in other subjects. As they
indicated in the interview responses, their
reply for how often they learn the use of
mathematics in other subjects was not
based on their classroom practice during
mathematics lessons but simply because
they get the application of mathematics in
other subjects. In the interview they
indicated that, during mathematics class,
they almost never learned the application of
the principles and laws of mathematics in
other subjects.
Thus the result implies that students’ real
life experiences were very rarely used in
the mathematics classroom learning
activities. It suggests that authentic
teaching of mathematics that connects
mathematical concepts, skills and strategies
to relevant real life contexts is missing. If
this dimension which is relevant for the
deeper understanding of mathematics and
transfer of learning is missing, students will
face challenge to apply their mathematical
knowledge and skills to a broad range of
real-world problems. Constructivists
suggested that students should find
personal relevance in their studies (Taylor,
Dawson, & Fraser, 1995) which means that
students will transfer the learning
experience provided in schools to the real
world (they will use skills developed in
schools to solve real-world problems) if
they are provided during classroom
learning with learning experiences that are
similar to the real world experiences.
A review by Chen (2003) indicated that
while the goal of education is to prepare
students to use their skills to solve real-
world problems, education is failing its
task. Chen’s review also indicated that the
common reason is that the learning
experience provided in schools is so
different from the experience in the real
world. Hence, it is obvious that students
couldn’t transfer learning into the work
world.
Teaching mathematics as abstractions using
symbols makes students lack interest in the
subject. This is because concepts and
theorems as abstractions are not observable
and hence difficult to learn. Students
consider their learning just as a useless
puzzle or playing a game using letters,
particularly x and y. However, students
need to know how the concepts and

strategies in mathematics relate with the
naturally occurring contexts. Not just
modeling concepts but the teachers are also
required how the concepts relate or applied.
Shared Control Scale
The mean value for the shared control scale
was (2.59). Students indicated that their
mathematics lessons are slightly more than
Seldom that they share control of their
learning with their teachers. This result is
consistent with previous studies (Aldridge,
Fraser, Taylor and Chen, 2000; Thao-Do,
Thi Bac-Ly and Yuenyong, 2016). It
suggests that students perceived that their
teachers were not sharing aspects of
learning mathematics with their students. In
other words, students showed that they
rarely had opportunities to be involved in
planning their learning including
articulating their learning objectives, the
design and management of learning
activities, in deciding what should be
included in their assessment and how they
should be assessed. If students have to
develop self regulation, to be accountable
for their learning, however they have to
practice planning about their learning and
achievement.
Particularly, in the interview students’
response confirmed that they never involve
in setting objectives, and rarely in
designing learning activities and in the
determination of assessment tasks. They
indicated that teachers also simply follow
the text books, demonstrate the examples
given in the texts, and give the text book
exercises as class works or as home works.
Students explained that their teachers never
share designing activities and usually
simply tell students the types of test
formats while the test taking dates arrive. If
shared control, an important dimension that
encourages students to take responsibility
for learning is missing, the very assumption
of active construction of knowledge will be
endangered.
The Uncertainty Scale
As indicated in the result section, the mean
score for the uncertainty scale is 2.77 and
this value was statistically significantly less
than the expected mean. This result is low
compared to previous studies (Aldridge,
Fraser, Taylor and Chen, 2000; Thao-Do,
Thi Bac-Ly and Yuenyong, 2016), in which
it was greater than 3. The current result
implies that it is seldom or sometimes that
students learn about the uncertain nature of
mathematics. That means students
indicated that they learn as if mathematics
is a universal endeavor that provide
accurate and objective knowledge of
reality. During interview students were
asked about their experiences of using
mathematics to obtain perfect answers to
problems including the geometrical
formulae. Students believed that
mathematics can provide perfect answers to
problems. They accepted the mathematical
rules (axioms) and theorems for
guaranteed. When this question is extended
to “can you find the exact area of a plot of
land which has an irregular shape?” They
started to hesitate and indicated that this is
not because mathematics failed to give
perfect answers rather it is due to the
irregular shape of the plots of land that it is
impossible to find the exact areas. They
were also asked that why positive numbers
are indicated on the right side of a number
line and the negative numbers to the left.
Their answer was because negative
numbers are smaller than positive numbers.
This response indicates that students
consider conventions or human agreements
as rules of mathematics.
Of the six items of the uncertainty scale,
students rated the two items “I learn how
the rules and theorems of mathematics
were invented” and “I learn that

mathematics is about creating rules and
theorems” relatively high. During the
interview students indicated that they
neither learned how the rules and theorems
were invented nor about creating new rules
and theorems. When asked why they rated
the two items high, students explained that
it is because they merely learn about
mathematical rules and about proving
theorems. But learning rules and theorems
is quite different from learning how the
rules and theorems were created or learning
how to create new ones. This type of
learning couldn’t help students to develop
mathematical models so that to tackle real-
world problems when they join the work
world.
The Critical Voice Scale
The Critical Voice Scale assess the extent a
classroom environment has been
established in which students feel that it is
legitimate to question the teachers
pedagogical plans and methods, and to
express concerns about any barrier to their
learning. The result showed that the mean
score is 3.06 which indicated that students
somehow express their concern about
barriers to their learning or understanding
of mathematics and it is similar to the result
for the study in Australia (Aldridge, Fraser,
Taylor and Chen, 2000). This mean value,
which is close to 3, suggests that teachers
are somehow accountable for their
pedagogical actions (instructional
activities).
It is common that teachers invite students
to ask any question concerning their
learning. Students have also indicated this
fact during the interview. They rather
pointed out that asking why they learn
every topic of the course is considered a
negative thing and that it is the teacher who
rarely tells why learning a certain topic is
so relevant. But students confirmed that it
is uncommon for them to challenge the
plans, methods and strategies teachers use
in the classroom. Hence, students’ response
for the scale is about their freedom to ask
what is not clear during learning, not
challenging the plans and methods of the
teacher. Interviewed students also agreed
that they know their freedom to ask what
they failed to understand otherwise they
never expected to comment on the
strategies of the teacher in his/her teaching.
In other words, students didn’t often
express their thoughts and criticisms about
their learning and how it might be
improved. Hence, this positive effect is not
great enough to change traditional
mathematics classrooms into highly
constructivist-oriented ones.
The Negotiation Scale
As has been displayed in the result section,
the greatest mean score was obtained for
the negotiation scale. This mean score,
3.75, showed that the learning environment
in mathematics classrooms emphasizes
student negotiation. Students perceived that
the classroom situation promotes student
interaction during mathematics learning.
They reported that the opportunity for
negotiation with their peers occurred often.
This result is greater than the results
reported in earlier studies (Aldridge, Fraser,
Taylor and Chen, 2000; Thao-Do, Thi Bac-
Ly and Yuenyong, 2016).
Group discussion has become to be
common in today’s classrooms. Some
educators are even arguing that active
learning has been equated with group
discussion (Lynch, 2010) and the
researcher feels that teachers are using
group discussion for topics which don’t
require divergent thinking. Hence, this
result might not be surprising and a rather
greater score may be expected for this
dimension. The interview data showed that
students are discussing just for the sake of

discussing not to develop learning from
each other. They indicated that discussion
is just to understand what is provided in the
textbook or by the teacher not to entertain
different opinions from each student or to
challenge what has been provided in the
text books.
Generally, the results for the personal
relevance, mathematical uncertainty and
shared control scales are low (between
seldom and sometimes) while the results
for critical voice and student negotiation
scales are modest. However, the interview
data showed that there is a need to improve
all results to actually create a learning
environment that is consistent with the
constructivist environment.
CONCLUSIONS
This mathematics classroom learning
environment study combined quantitative
and qualitative methods in Benishangu
Gumuz and Amhara region secondary
schools. Students perceived more critical
voice and negotiations with peers and less
personal relevance, mathematical
uncertainty and shared control. In other
words, there is a need to promote
constructivist-oriented teaching in school
classrooms, especially in terms of students’
perceptions of shared control, the relevance
of teaching and the uncertainty of
mathematics. Overall, secondary school
students in Benishangul Gumuz and
Amhara regions perceived their current
mathematics classroom learning
environments as modest.
Although it was not the purpose of this
study, the instrument validation process
showed that the questionnaire exhibited
good factorial validity and internal
consistency reliability and thus can be used
in further classroom learning environment
studies. That is the questionnaire can serve
as a useful means of evaluating the degree
to which students felt that the principles of
constructivism had been implemented in
the mathematics classes.
RECOMMENDATIONS
Constructivism (particularly social
constructivism theory) sees mathematics
primarily as a social construct, as a product
of culture, subject to correction and change.
Social constructivists argue that
mathematics is in fact grounded by much
uncertainty. Like the other sciences,
mathematics is viewed as an empirical
endeavor whose results are constantly
evaluated and may be discarded (Ernest,
2004). On the other hand, our students have
evidenced that these are not usually
practiced in their mathematics classrooms.
Hence the results signal that there is a need
to orient teachers with the constructivist
educational theory and its classroom
application.
The findings in this research revealed that
mathematics teachers seldom or sometimes
provide classroom learning contexts that
makes content relevant to students’ lives,
connecting mathematics to students' out-of-
school experiences and making use of
students' everyday experiences as a
meaningful context for the development of
students' mathematical knowledge. Hence,
teachers should provide task that are related
to students’ everyday life. Shared control
could be improved through the
development of formative assessment
which allows students to identify and select
their assignment and project topics
themselves, and hence play a larger role in
planning for their learning. Hence, teacher
development programs should focus on
such important constructivist principles.
Moreover, teachers should be aware that
they have to devise mechanisms to bring
the real world into the classroom and

integrate mathematics into authentic
learning situations.
REFERENCES
Aldridge, Jill M., Fraser, Barry J., Taylor,
Peter C. & Chen, Chung-Chih.
(2000).Constructivist Learning
Environments in a Cross-national
Study in Taiwan and Australia.
International Journal of Science
Education. 22, 37-55.
Chen, Catherine. (2003). A constructivist
approach to teaching:
Implications in teaching
computer networking.
Information Technology,
Learning, and Performance
Journal. 21(2), 17-26.
Cooper, Peter A. (1993). Paradigm shifts in
instruction: From behaviorism to
cognitivism to constructivism.
Educational Technology. 33(5),
12-19.
Dryden, M. & Fraser, B.J. (1996,
April). Use of classroom
environment instruments in
monitoring urban systemic reform.
Paper presented at the annual
meeting of the American
Educational Research Association,
New York.
Dryden, M. & Fraser, B.J. (1998,
April). Awareness versus
acceptance of systemic change
among secondary science
teachers. Paper presented at the
annual meeting of the National
Association for Research in
Science Teaching, San Diego, CA.
Ernest, Paul. (2004). The Philosophy of
Mathematics Education.
Routledge Falmer, Taylor
& Fransis Group.
Fraser, Barry J., Dryden, Michael, &
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Lynch, Dana N. (2010). Student-Centered
Learning: The Approach That
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Ministry of Education. (2010). Education
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Nix, Rebekah K., Fraser, Barry J., &
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American Educational Research
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(1995, April). A constructivist
perspective on monitoring
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under transformation. Paper
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Fransisco, CA.
Taylor, P.C., Fraser, B.J. & White, L.R.
(1994). The revised CLES: A
questionnaire for educators
interested in the constructivist
reform of school science and
mathematics. Paper presented at
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Taylor, PC. & Fraser BJ. (1991).
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American Educational Research
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Thao-Do, Thi Phuong, Thi Bac-Ly, Dang
& Yuenyong, Chokchai. (2016).
Learning Environment in
Vietnamese Physics Teacher
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Study of a State University in
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